Editing Analysis of variance (section)

== Classes of models ==
There are three classes of models used in the analysis of variance, and these are outlined here.

===Fixed-effects models===
{{Main|Fixed effects model}}
The fixed-effects model (class I) of analysis of variance applies to situations in which the experimenter applies one or more treatments to the subjects of the experiment to see whether the [[response variable]] values change. This allows the experimenter to estimate the ranges of response variable values that the treatment would generate in the population as a whole.
[[File:Fixed_effects_vs_Random_effects.pdf|thumb|291x291px|Fixed effects vs Random effects]]

===Random-effects models===
{{Main|Random effects model}}
Random-effects model (class II) is used when the treatments are not fixed. This occurs when the various factor levels are sampled from a larger population. Because the levels themselves are [[random variable]]s, some assumptions and the method of contrasting the treatments (a multi-variable generalization of simple differences) differ from the fixed-effects model.<ref>Montgomery (2001, Chapter 12: Experiments with random factors)</ref>

===Mixed-effects models===
{{Main|Mixed model}}
A mixed-effects model (class III) contains experimental factors of both fixed and random-effects types, with appropriately different interpretations and analysis for the two types.

===Example===
Teaching experiments could be performed by a college or university department to find a good introductory textbook, with each text considered a treatment. The fixed-effects model would compare a list of candidate texts. The random-effects model would determine whether important differences exist among a list of randomly selected texts. The mixed-effects model would compare the (fixed) incumbent texts to randomly selected alternatives.

Defining fixed and random effects has proven elusive, with multiple competing definitions.<ref>Gelman (2005, pp. 20–21)</ref>